Computational aspects of minimizing conditional value-at-risk

Abstract

Abstract.We consider optimization problems for minimizing conditional valueat-risk (CVaR) from a computational point of view, with an emphasis on financial applications. As a general solution approach, we suggest to reformulate these CVaR optimization problems as two-stage recourse problems of stochastic programming. Specializing the L-shaped method leads to a new algorithm for minimizing conditional value-at-risk. We implemented the algorithm as the solver CVaRMin. For illustrating the performance of this algorithm, we present some comparative computational results with two kinds of test problems. Firstly, we consider portfolio optimization problems with 5 random variables. Such problems involving conditional value at risk play an important role in financial risk management. Therefore, besides testing the performance of the proposed algorithm, we also present computational results of interest in finance. Secondly, with the explicit aim of testing algorithm performance, we also present comparative computational results with randomly generated test problems involving 50 random variables. In all our tests, the experimental solver, based on the new approach, outperformed by at least one order of magnitude all general-purpose solvers, with an accuracy of solution being in the same range as that with the LP solvers.

Multiperiod Stochastic Optimization Problems with Time-Consistent Risk Constraints

“…For an overview of the many developments since then, see the monographs [10,17,20] and the survey [16]. In particular, the example of the coherent risk measure Conditional-Value-At-Risk 1 (CVaR) is suitable for applications, mainly because of its reformulation as a linear optimization problem [21], leading to efficient implementations for mean-risk problems [12,13].…”

CVaR minimization by the SRA algorithm

“…This model can be formulated as a two-stage stochastic programming problem (see Künzi-Bay and Mayer, 2006):…”

“…Rockafellar and Uryasev (2000) formulate this model as a linear programming problem (LP). Künzi-Bay and Mayer (2006) model the portfolio optimization as a two-stage stochastic programming problem. So we can handle the CV aR portfolio optimization model with algorithms designed to solve two-stage problems.…”

An algorithm for sequential tail value at risk for path-independent payoffs in a binomial tree

“…Fishburn (1964), Föllmer and Schied (2004), Dentcheva and Ruszczyński (2003). Considerable effort has been put in efficient algorithms that can cope with the huge amount of restrictions in the corresponding LP-problems, in particular by means of cutting-plane methods, see Klein Haneveld and van der Vlerk (2006), Künzi-Bay and Mayer (2006), Rudolf and Ruszczyński (2008), Luedtke (2008), Fábián et al (2009).…”